scholarly journals ON AUTOMORPHISMS OF GENERALIZED CUNTZ ALGEBRAS

1998 ◽  
Vol 09 (04) ◽  
pp. 493-512 ◽  
Author(s):  
YOSHIKAZU KATAYAMA ◽  
HIROAKI TAKEHANA
Keyword(s):  
Fixed Point ◽  
Finite Type ◽  
Invertible Operator ◽  
Cuntz Algebra ◽  
Gauge Action ◽  
Cuntz Algebras ◽  
C Algebra ◽  
Fixed Point Algebra

Let X be a full right Hilbert B-bimodule of finite type and [Formula: see text] be its generalized Cuntz algebra. We give a notion that the C*-algebra B is X-aperiodic. We show that the fixed point algebra ℱX for a gauge action is simple if and only if the C*-algebra B is X-aperiodic. For a invertible operator U on X with some properties, a quasi-free automorphism αU of [Formula: see text] is defined. We give some conditions in order that αU is inner in the case that B is X-aperiodic. We apply them to the automorphism αU on Cuntz–Krieger algebras.

scholarly journals On pathological properties of fixed point algebras in Kirchberg algebras

2019 ◽  
Vol 150 (6) ◽  
pp. 3087-3096
Author(s):  
Yuhei Suzuki
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Approximation Property ◽  
Infinite Group ◽  
Cuntz Algebra ◽  
C Algebra ◽  
Outer Action ◽  
Reduced Group ◽  
Fixed Point Algebra

AbstractWe investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

scholarly journals A Short Note on the Continuous Rokhlin Property and the Universal Coefficient Theorem in E-theory

2015 ◽  
Vol 58 (2) ◽  
pp. 374-380 ◽  
Author(s):  
Gábor Szabó
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Short Note ◽  
Circle Actions ◽  
Continuous Action ◽  
C Algebra ◽  
C Algebras ◽  
Group A ◽  
Fixed Point Algebra ◽  
Universal Coefficient Theorem

AbstractLet G be a metrizable compact group, A a separable C*-algebra, and α:G → Aut(A) a strongly continuous action. Provided that α satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in E-theory passes from Ato the crossed product C*-algebra A⋊α G and the ûxed point algebra Aα. This extends a similar result by Gardella for KK-theory in the case of unital C*-algebras but with a shorter and less technical proof. For circle actions on separable unital C*-algebras with the continuous Rokhlin property, we establish a connection between the Etheory equivalence class of A and that of its fixed point algebra Aα.

Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra On

2009 ◽  
Vol 6 (2) ◽  
pp. 339-380 ◽  
Author(s):  
A.L. Carey ◽  
J. Phillips ◽  
A. Rennie
Keyword(s):  
Automorphism Group ◽  
General Theory ◽  
Index Theory ◽  
Index Theorem ◽  
Cuntz Algebra ◽  
Spectral Flow ◽  
Gauge Invariant ◽  
Gauge Action ◽  
Cuntz Algebras ◽  
Modular Automorphism

AbstractThis paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K1-group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.

scholarly journals An action of the Klein four-group on the irrational rotation C*-algebra

1997 ◽  
Vol 56 (1) ◽  
pp. 135-148
Author(s):  
P.J. Stacey
Keyword(s):  
Fixed Point ◽  
Irrational Rotation ◽  
C Algebra ◽  
Fixed Point Algebra

Explicit automorphisms of the irrational rotation algebra are constructed which are associated with the two 2 × 2 diagonal integer matrices of determinant −1. The fixed point algebra of the product of these two automorphisms is shown to be isomorphic to the fixed point algebra of the flip.

scholarly journals SATURATED ACTIONS BY FINITE-DIMENSIONAL HOPF *-ALGEBRAS ON C*-ALGEBRAS

2008 ◽  
Vol 19 (02) ◽  
pp. 125-144 ◽  
Author(s):  
JA A. JEONG ◽  
GI HYUN PARK
Keyword(s):  
Finite Group ◽  
Finite Type ◽  
Group Action ◽  
Hopf Algebras ◽  
Equivalent Condition ◽  
Finite Group Action ◽  
C Algebra ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
A Finite Group

If a finite group action α on a unital C*-algebra M is saturated, the canonical conditional expectation E : M → Mα onto the fixed point algebra is known to be of index finite type with Index (E) = |G| in the sense of Watatani. More generally, if a finite-dimensional Hopf *-algebra A acts on M and the action is saturated, the same is true with Index (E) = dim (A). In this paper, we prove that the converse is true. Especially in case M is a commutative C*-algebra C(X) and α is a finite group action, we give an equivalent condition in order that the expectation E : C(X) → C(X)α is of index finite type, from which we obtain that α is saturated if and only if G acts freely on X. Actions by compact groups are also considered to show that the gauge action γ on a graph C*-algebra C*(E) associated with a locally finite directed graph E is saturated.

scholarly journals The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

2021 ◽  
pp. 1-52
Author(s):  
M. ALI ASADI-VASFI ◽  
NASSER GOLESTANI ◽  
N. CHRISTOPHER PHILLIPS
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Positive Part ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Cuntz Semigroup ◽  
Fixed Point Algebra ◽  
Stably Finite ◽  
A Finite Group

Abstract Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha \colon G \to {\text{Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let $A^{\alpha}$ be the fixed point algebra. Then the radius of comparison satisfies ${\text{rc}} (A^{\alpha }) \leq {\text{rc}} (A)$ and ${\text{rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{\text{card} (G))} \cdot {\text{rc}} (A)$ . The inclusion of $A^{\alpha }$ in A induces an isomorphism from the purely positive part of the Cuntz semigroup ${\text{Cu}} (A^{\alpha })$ to the fixed points of the purely positive part of ${\text{Cu}} (A)$ , and the purely positive part of ${\text{Cu}} ( C^* (G, A, \alpha ) )$ is isomorphic to this semigroup. We construct an example in which $G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$ , A is a simple unital AH algebra, $\alpha $ has the Rokhlin property, ${\text{rc}} (A)> 0$ , ${\text{rc}} (A^{\alpha }) = {\text{rc}} (A)$ , and ${\text{rc}} (C^* (G, A, \alpha ) = ( {1}/{2}) {\text{rc}} (A)$ .

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

Morita equivalences between fixed point algebras and crossed products

1999 ◽  
Vol 125 (1) ◽  
pp. 43-52 ◽  
Author(s):  
CHI-KEUNG NG
Keyword(s):  
Fixed Point ◽  
Quantum Group ◽  
Type Theorem ◽  
Compact Quantum Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Morita Equivalent ◽  
Fixed Point Algebra

In this paper, we will prove that if A is a C*-algebra with an effective coaction ε by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras.

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